Mixing
Question on exclusive or vs inclusive or
Here's what my textbook asks me to prove:
From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.
I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.
I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.
logic
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
add a comment |
Here's what my textbook asks me to prove:
From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.
I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.
I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.
logic
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 '18 at 10:31
@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 '18 at 10:34
I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 '18 at 17:37
add a comment |
Here's what my textbook asks me to prove:
From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.
I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.
I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.
logic
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
Here's what my textbook asks me to prove:
From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.
I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.
I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.
logic
logic
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
asked Nov 22 '18 at 10:26
Ashish KAshish K
815613
815613
How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 '18 at 10:31
@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 '18 at 10:34
I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 '18 at 17:37
add a comment |
How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 '18 at 10:31
@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 '18 at 10:34
I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 '18 at 17:37
How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 '18 at 10:31
How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 '18 at 10:31
@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 '18 at 10:34
@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 '18 at 10:34
I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 '18 at 17:37
I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 '18 at 17:37
add a comment |
3 Answers
3
active
oldest
votes
Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.
Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
add a comment |
"if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
add a comment |
The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
is not problematic.
for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.
Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
add a comment |
Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.
Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
add a comment |
Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.
Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.
Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
answered Nov 22 '18 at 10:38
TaroccoesbroccoTaroccoesbrocco
5,14761839
5,14761839
add a comment |
add a comment |
"if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
add a comment |
"if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
add a comment |
"if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
"if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
answered Nov 22 '18 at 10:32
Vee Hua ZhiVee Hua Zhi
759224
759224
add a comment |
add a comment |
The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
is not problematic.
for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
add a comment |
The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
is not problematic.
for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
add a comment |
The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
is not problematic.
for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
is not problematic.
for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
answered Nov 22 '18 at 10:38
Mohammad Riazi-KermaniMohammad Riazi-Kermani
41.3k42060
41.3k42060
add a comment |
add a comment |
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How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 '18 at 10:31
@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 '18 at 10:34
I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 '18 at 17:37